Abstract

We study the problem of bifurcation of critical periods of a time-reversible polynomial system of degree . We first present a new method to find the number of zeros of the period function. Then applying our results, we study the number of critical periods for some polynomial systems and obtain new results.

1. Introduction

Consider a two-dimensional analytic real differential system of the form
where we suppose there is a nondegenerate center at the origin . Let denote the orbit passing through the point with of (1) and denote its period. As we all know, the isolated zeros of derivative of are named critical periods. An interesting problem is to investigate the number of the critical periods. This is an important problem in the research of period functions and a lot of results have been obtained for polynomial differential systems, for example, monotonicity [1–8], finiteness of critical periods [9, 10] and isochronicity [11–13], and local bifurcation of critical periods [14]. The number of critical periods was also discussed in [15–17] for perturbations of isochronous vector fields.

Recently, the authors in [18] studied the following system:
where , ,
Under the conditions above, system (2) is time-reversible and hence has a center at the origin.

As shown in [15–20], for , the period function of system (2) can be written as
where is a positive constant. The authors [18] gave expressions of and . It is also proved in [18] that one critical period can appear for the case and two critical periods can be found for the case .

In this paper, we consider the following system:
where
, , , , and both and are small parameters. From the above expressions, we know that the system (5) satisfies the following properties: (i) it is a time-reversible system, and the perturbations with are of general form; (ii) for all , the origin is a center, and for , the origin is isochronous since there exist polynomials and such that , , , .

Let denote the period of the periodic orbit of system (5) passing through the point . Then it has the following expansion:
where
and since (5) becomes (2) as , we have .

Next section, we give some preliminary lemmas.

2. Preliminary Lemmas

Lemma 2 (see [18]). Let and in (5). Then
where and are constants depending on the coefficients appearing in , , , and . Furthermore, for sufficiently small ,(i)the center preserves the isochronicity when ; period function is increasing (resp., decreasing) for when and (resp., 0);(ii)there is at most one critical period in when . Moreover, there is exactly one critical period in if and only if

Lemma 3 (see [18]). Let and in (5). Then
where and , are constants depending on the coefficients appearing in , , , and . Furthermore, for sufficiently small , one has the following results.(i)The center preserves the isochronicity when
(ii)If (20) do not hold, there are at most two critical periods in and the maximum is achievable.

3. Main Results

As in [18], one can make the polar coordinates , , so that system (5) becomes
where and are given by (13) and (14),
Obviously, when , which implies that the unperturbed system (5) has an isochronous center at which is called a rigidly or uniformly isochronous center. We have the following fundamental result.

Proof. We follow the idea of proving Lemma 1 which is Theorem 2.1 given in [18].From (21), we have
Let be the solution of the above equation satisfying the initial condition . It can be written as a series of
where , , .Substituting (30) into differential equation (29) and comparing the coefficients of , we have
We can write , where , , . Then, substituting into (31), then comparing the coefficients of and , we can obtain
Solving the above ODEs associated with the initial values and , we can get (27) and (28). By (21) we have
Therefore,
It implies that
Thus, (26) is proved. This ends the proof of Theorem 4.

With the same method as for , we can compute for and give an expression of for from the proof of Theorem 4, which are omitted here.

Define

Another fundamental result is as follows.

Theorem 5. Let be defined as before with . Then for , one has the following. (i) The period function of system (5) is increasing (resp., decreasing) in if (resp., 0) for all . (ii) If is not identically zero, the number of critical periods of in is not more than the number of zeros (take the multiplicity into consideration) of in . And there are exactly critical periods if has exactly simple positive zeros. (iii) If , the number of critical periods of in is not larger than the number of zeros of in . And critical periods can appear if has simple zeros. Similarly, the period function of (5) is increasing (resp., decreasing) in if (resp., < 0).

Proof. From (8) and (9)
If , , then there exists a such that for , has the same sign with . For , the sign of the function is the same as . Then the conclusion (i) is proved.Further, suppose that are the zeros of with the multiplicity , respectively. We only need to prove that the number of critical periods of is less than or equal to . For the purpose, it suffices to prove that has at most roots. Thus, we only need to prove that has at most zeros near , . At first, we prove that has at most zeros near for small. If it is not the case, then has at least zeros near ; that is, there exists such that has at least zeros near , where . By Rolls theorem, has at least zeros near , and has at least one zero near . Thus, . Letting , we have . This is a contradiction. Thus, has at most zeros near . Using the same method, for sufficiently small , or has at most zeros near . Thus, has at most critical periods for each . Thus, we have proved the first part of (ii).For the second part of (ii), assume that are the simple zeros of ; that is, and , . Then
By the Implicit Function Theorem, there is a unique function such that and , . Therefore are simple zeros of . By the same method, there is a unique function such that , and . The second part of (ii) is proved.When , then
Then, we can prove conclusion (iii) in the same way as proving conclusion (ii). The proof is completed.

4. Application

In this section, we apply Theorems 4 and 5 to the cases , ; , , and , , respectively.

Case 1 (, ). In this case, system (5) becomes
where . For system (40), is a center because of the symmetry.We only need to consider the case of in (40); otherwise, we can use the transformation to change (40) into the same form with opposite signs to the coefficients. Further, suppose , otherwise, system (40) can be simplified as the form of system (41) by the transformation , . In this case, system (40) becomes
It is easy to conclude that system (41) has a first integral of the form

Theorem 6. For system (41), one has (i)where . For sufficiently small and , there is at most one critical period if is not identically zero and ; there is exactly one critical period if and only if . Otherwise, the period function is increasing for (resp., decreasing) if (resp., < 0); (ii) if , then for ,
where
There can appear three critical periods in if is not identically zero.

Proof. For system (41), from (13), (14), (22)–(25), we have
From (27), we have
From (10), we have
Then, from [18], we have
where and
Support that . By (50) we have four zeros of as follows:
where is given by . It is not difficult to examine that is not in and is not in . In addition, if and only if . Thus, there is at most one critical period in . If , has no zero in . From the expression of , we can easily check that (resp., 0) when (resp., 0). Therefore, for sufficiently small and , (resp., 0) if (resp., 0). The proof of conclusion (i) is completed.For conclusion (ii), let , which gives or . From (26), we have that
Then we expand at by letting
By (52), we have
Note that
and that
Therefore, can be taken as free parameters. Hence, we can change the sign of , satisfying
which ensures that has three positive zeros in near . Thus, conclusion (ii) is proved.

We remark that for cubic system (40), we obtain three critical periods using , which is one more than the results in Lemma 2.3 obtained in [18].

Case 2 (, ). In this case, system (40) becomes
where we suppose . This is a new system which is not studied in [18]. For the case , by the rescaling
system (58) can be written as
where
If , system (58) can be simplified as (60) similarly by the change , .

Theorem 7. For system (60), one has (i)where . If is not identically zero, there are at most two critical periods in for sufficiently small and , and the maximum can be achievable; (ii) if , one has
and there are at most two critical periods in if is not identically zero.

Proof. From (13), (14), (22)–(25), we have that for system (60)
From (27) and (28), we have
Hence, from (10),
Thus,